Understanding the strategies
To understand how ant queens might fare in this evolutionary game, imagine two S. invicta queens ready to leave their home nests. One queen let's call her Mono carries a gene that causes her to leave her home nest and start a colony by herself. So Mono always has a "go it alone" strategy she doesn't get any help from other ants, but she doesn't have to share her resources with them either.
The other queen let's call her Poly carries a gene that causes her to try to stay in her own nest and reproduce alongside her mother or to join another colony and reproduce alongside the queen there. So Poly always has a "cooperative" strategy she welcomes the help of other ants and shares her resources with them.
The "go it alone" strategy is a great one if you can make it work. If it works and the queen does manage to start a new colony, she will get a bonanza of genes in the next generation the entire colony (and any queens it produces) will be the founder's direct descendents. However, if the "go it alone" strategy fails, it will be a complete flop. A queen that tries to start a colony by herself and fails is likely to die without getting a single gene into the next generation. The "cooperative" strategy is a safer bet joining another nest is pretty likely to yield some genes in the next generation, just not a "bonanza" of genes.
Situation 1 Sparse population
Imagine that Mono and Poly live in an area with a lot of unoccupied nest sites available (see graphic below). In this situation, Mono would "win" she would start a new nest, and the entire colony would end up carrying her genes. Mono's "go it alone" strategy works well when there are enough resources for loners to build a colony from scratch. Poly would "lose" in this situation because as one of several queens in a nest she wouldn't have the resources to reproduce as much as Mono. Poly's "cooperative" strategy backfires here she shares her resources with others, even though there is plenty to go around.
Situation 2 Dense Population
On the other hand, if Mono and Poly live in an area where the good nest sites are already taken (see graphic below), Poly would "win" and Mono would "lose." Mono's genes would cause her to try to start a colony by herself but because space is limited, she wouldn't be able to find a nest site and would die without leaving any offspring. "Going it alone" is likely to be a complete failure when things are crowded very, very few solo queens would be able to successfully start a nest. However, Poly's "cooperative" strategy works great in this situation. She doesn't have to find the space and resources to start a whole new colony she just needs to find a colony willing to accept her and share some resources either her own or a closely related nest. Poly would get to reproduce and would get some of her genes into the next generation. Cooperation pays when times are tough and space and resources are limited.
So for S. invicta queens, the winning strategy for getting genes into the next generation depends on whether or not good nest sites are available: with nest sites available, start a colony by yourself; with very few available nest sites, join other queens in a multiple-queen colony.